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QUESTION

According to David Wells ({\em The Penguin dictionary of curious
and interesting numbers})

$$4^4+3^3+8^8+5^5+7^7+9^9+0^0+8^8+8^8=438,579,088.$$

Do you agree?

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ANSWER

Many students will, no doubt, do the same as I did when I first
saw this problem: type the left hand side of the equation into
their calculator without thinking, so getting an error message!

The alledged result is true only if one adopts the convention that
$0^0=0$. But the function $z=x^y$ has an essential singularity at
$(x,y)=(0,0)$ so it is not possible to assign a reasonable value
to $0^0$. One might adopt the convention that
$0^0=\lim_{x\to0+}x^x$ but this gives 1 not 0.

The question is designed to get students to think for themselves.




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